Cartan Subalgebras of gl∞
Cartan Subalgebras of gl∞
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gl∞ 的嘉当子代数
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通讯作者：
I. Penkov
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作者：
K. Neeb;I. Penkov
设V是特征为零的域K上的向量空间，V是V上分离V的点的线性泛函空间。设V <$V <$是V上有限秩算子的李代数，设gl（V，V <$）：= V <$V <$.定义gl（V，V_n）的一个Cartan子代数为每个元素都是半单的极大子代数的中心化子，然后在K是代数闭的假设下给出gl（V，V_n）的所有Cartan子代数的如下刻划. gl（V，V（？）是Cartan子代数当且仅当它等于？j（V j（V j））（V 0 <$V 0 <$）对于某些一维子空间V j <$V和（V j）<$$> V <$，其中（Vi）<$（V j）= δi j K，并且使得空间V 0 <$=<$j（V j）<$$> V <$和V 0 =<$j（（V j）<$）<$$> V满足V 0 <$（V 0）= {0}。然后，我们讨论子空间V j和（V j）的显式构造。我们的第二个主要结果是gl（V，V）的Cartan子代数可以被描述为伴随表示局部有限的局部幂零自正规子代数，或者被描述为与gl（V，V）的极大局部幂零h-子模重合的子代数h，使得h的伴随表示局部有限。引言这是一个有趣的问题，一类子代数的无限维李代数，在一个领域K的特征零，发挥类似的作用，嘉当子代数的有限维李代数。尽管无限维李代数在过去的30年里得到了广泛的研究，但这个问题还没有明确的答案。最好理解的情况是Kac-Moody代数和扩展仿射李代数（参见[BP 95]，[PK 83]，[AABGP 97]和其中的参考文献），其特殊之处在于它们的Cartan子代数是有限维的。无限维李代数的Cartan子代数不再是有限维的最简单的例子是K中有1/4个非零元素的无限矩阵的李代数gl∞，在文献中没有系统地研究gl∞的所有Cartan子代数。本文的目的是填补gl∞和下面定义的更大类李代数gl（V，V ∞）的这一空白。有限维李代数g的Cartan子代数h的以下三种定义是等价的：（C1）h是局部幂零自正规化子代数;（C2）h与g的极大局部幂零h-子模一致，即，h = g（h），其中g（h）= { x ∈ g：（n ∈ N）（ad h）（x）= {0} } ; 2002年12月4日由编辑接收; 2003年3月13日修订。第一作者的工作得到了加州大学滨江分校和DFG赠款的部分支持。第二作者的工作部分得到了MSRI的支持，由NSF拨款，并由波恩的马克斯普朗克数学研究所提供。AMS科目分类：小学：17 B65;中学：17 B20。c ©加拿大数学学会，2003年。
Let V be a vector space over a field K of characteristic zero and V∗ be a space of linear functionals on V which separate the points of V . We consider V ⊗ V∗ as a Lie algebra of finite rank operators on V , and set gl(V,V∗) := V ⊗ V∗. We define a Cartan subalgebra of gl(V,V∗) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of gl(V,V∗) under the assumption that K is algebraically closed. A subalgebra of gl(V,V∗) is a Cartan subalgebra if and only if it equals ⊕ j ( V j ⊗ (V j )∗ ) ⊕ (V 0 ⊗V 0 ∗) for some one-dimensional subspaces V j ⊆ V and (V j )∗ ⊆ V∗ with (Vi )∗(V j ) = δi j K and such that the spaces V 0 ∗ = ⋂ j (V j ) ⊥ ⊆ V∗ and V 0 = ⋂ j ( (V j )∗ )⊥ ⊆ V satisfy V 0 ∗(V 0) = {0}. We then discuss explicit constructions of subspaces V j and (V j )∗ as above. Our second main result claims that a Cartan subalgebra of gl(V,V∗) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra h which coincides with the maximal locally nilpotent h-submodule of gl(V,V∗), and such that the adjoint representation of h is locally finite. Introduction It is an interesting question which class of subalgebras of an infinite-dimensional Lie algebra, over a field K of characteristic zero, play a role similar to Cartan subalgebras of a finite-dimensional Lie algebra. Despite the fact that infinite-dimensional Lie algebras have been studied extensively in the last 30 years, there is no definitive answer to this question. The best understood cases are those of Kac-Moody algebras and extended affine Lie algebras (see [BP95], [PK83], [AABGP97] and the references therein), whose specific is that their Cartan subalgebras are finite-dimensional. The simplest example of an infinite-dimensional Lie algebra whose Cartan subalgebras are no longer finite-dimensional is the Lie algebra gl∞ of infinite matrices with finitely many non-zero entries in K, and in the literature there is no systematic investigation of all Cartan subalgebras of gl∞. The purpose of the present paper is to fill in this gap for gl∞ and for the larger class of Lie algebras gl(V,V∗) defined below. The following three definitions of a Cartan subalgebra h of a finite-dimensional Lie algebra g are equivalent: (C1) h is a locally nilpotent self-normalizing subalgebra; (C2) h coincides with the maximal locally nilpotent h-submodule of g, i.e., h = g(h), where g(h) = { x ∈ g : (∃n ∈ N) (ad h)(x) = {0} } ; Received by the editors December 4, 2002; revised March 13, 2003. The first author’s work was supported in part by the University of California at Riverside and by a grant of the DFG. The second author’s work was supported in part by MSRI, by an NSF grant, and by the Max Planck Institut für Mathematik in Bonn. AMS subject classification: Primary: 17B65; secondary: 17B20. c ©Canadian Mathematical Society 2003.